पाटीगणित (श्रीधराचार्य - प्राचीन संस्कृत टीका एवं प्रो. कृपाशंकर शुक्ल सविस्तार व्याख्या)

CUBE 11
(7) Cube (ghana)
Rules for cubing :
27-28. [Let the last digit of the given number be called
the ‘last’ (antya) and the last-but-one digit the ‘first’ (âdi) or
the ‘preceding’ (pûrva).]
Set down the cube of the ‘last’; then set down,
(successively) one place forward (sthânâdhikyaṁ), (i) the square
of the ‘last’ as multiplied by thrice the ‘preceding’, (ii) the
square of the ‘first’ as multiplied by the ‘last’ as well as by 3,
and (iii) the cube of the ‘first’. This gives the cube of the
combined number (formed by the ‘last’ and the ‘first’) (niryukta-
râśi), which should now be treated as the ‘last’ (provided there
be more than two digits in the given number).¹
The (continued) product of three equal quantities ;² or,
in the series having 1 for the first term and common difference
(considering the last two terms, designating them as the ‘first’
or ‘preceding’ and the ‘last’ respectively), the last multiplied
by thrice the ‘first’, and increased by 1, and that added to the
cube of the ‘preceding’, is also the cube.³
Of the three rules stated here, the first one is the main method of
cubing a number. To illustrate it by an example, let us find the
cube of 256.
To begin with we treat 2 as the ‘last’ and 5 as the ‘first’. Then
writing the cube of the ‘last’ (i.e., 2³, or 8), then in the next place the
square of the ‘last’ as multiplied by thrice the ‘first’ (i.e., 3×5×2², or 60),
then in the next place the square of the ‘first’ as multiplied by thrice the
‘last’ (i.e., 3×2×5², or 150), and then in the next place the cube of the
‘first’ (i.e., 5³, or 125), we have
0 0 0 0
8
6 0
1 5 0
1 2 5

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¹ Cf. BrSpSi, xii. 6 ; GSS, ii. 47 ; GT, p. 11, vs. 25 ; L (ASS), p. 23,
vs. 24-25(i) ; GK, I, p. 7, lines 17-19, p. 8, lines 1-2.
² Cf. Ā, ii. 3(ii) ; BrSpSi, xii. 62(ii) ; GSS, ii. 43(i) ; MSi, xv. 6(i) ;
GT, p. 11, line 10 ; SiŚe, xiii. 4(i) ; L (ASS), p. 23, vs. 24(i) ; GK, I, p. 7,
line 16.
³ Cf. GT, p. 11, lines 7-9 ; GK, I, p. 8, lines 3-4.